Group completion

Given a topological monoid $M$, the group completion is the space $\Omega BM$, where $BM$ is the classifying space of $M$ (thinking of $M$ as a topological category). If $M$ is already a topological group, then this operation does not change $M$ up to homotopy equivalence. Under some assumptions, we have the following description of the homology of the group completion. If we treat the monoid ${\pi }_0(M)$ as a directed system (with maps given by the monoid operation), then

$$\underset{\alpha \in {\pi }_0(M)}{\lim }{H}_{\ast }(M_{\alpha })=H_{\ast }((\Omega BM{)}_0)$$

in situations where the direct limit on the left-hand side is well-defined. In particular, if we consider the monoid ${⨿}_{g}B{\Gamma }_{g,2}$, the homology of the group completion is precisely the stable homology. The plus construction can often be used to give an alternative construction of the group completion.